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Unformatted text preview: BOOLEAN RELATION THEORY AND MORE by Harvey M. Friedman Ohio State University MIT LOGIC SEMINAR November 3, 2009 friedman@math.ohio-state.edu www.math.ohio-state.edu/~friedman 4:30-6PM We will start with the MORE. We present the Unprovable Upper Shift Fixed Point Theorem. It asserts the existence of relations on Q - and in fact, arithmetical relations on Q. It is provable from large cardinals but not in ZFC. It is provably equivalent to consistency of a large cardinal axiom system, and so is, at least indirectly, 01. It has simple finite forms that are explicitly 02 and 01. It has a natural Template that supports a general theory. SOME EASY DEFINITIONS We use Q for the rationals. We say that x,y Q k are order equivalent iff for all 1 i,j k, x i &lt; x j y i &lt; y j . We say that A Q k is order invariant iff for all order equivalent x,y Q k , x A y A. We say that R Q k Q k is order invariant iff R is order invariant as a subset of Q 2k . We say that R Q k Q k is strictly dominating iff for all x,y Q k , R(x,y) max(x) &lt; max(y). Write SDOI(Q k ,Q k ) for the set of all strictly dominating order invariant R Q k Q k . We define R[A] for A Q k , as {y: ( x Q k )(R(x,y))}. For x Q k , define the upper shift us(x) to be the result of adding 1 to every nonnegative coordinate of x. For A Q k , let us(A) = {us(x): x A}. Write cube(A,0) for the least B k such that A B k 0 B. UNPROVABLE UPPER SHIFT FIXED POINT THEOREM. For all R SDOI(Q k ,Q k ), some A = cube(A,0)\R[A] contains us(A). Relevant large cardinals required most memorably stated in terms of the k-SRP = order k stationary Ramsey property. is k-SRP iff is a limit ordinal where every partition of the unordered k-tuples from into two pieces has a stationary homogenous set. For k 2, the least k-SRP is strongly inaccessible, even weakly compact, even totally indescribable, etc. But below . Stationary Ramsey property hierarchy. Subtle cardinal hierarchy. Almost ineffable hierarchy. Ineffable hierarchy. All provably intertwined. See H. Friedman, Subtle Cardinals and Linear Orderings, Annals of Pure and Applied Logic 107 (2001), 1-34....
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